User:PMarmottant/Mixing

Mixing molecules in a solute is a crucial operation for biological analysis or chemical reactions. We will see in this chapter a specificity of microfluidics: mixing is long.

Origin of diffusion in a liquid: brownian motion
In a liquid all molecules are agitated due to their temperature. A big molecule embedded in a liquid is thus subject to continuous water impacts by neighbour molecules. This impacts are random and produces step motion in random directions.

To understand the effect of this random motion, we consider a 1D model (along one axis) with random steps of finite size $$\ell$$, that are occurring periodically in time with period $$\Delta t$$.

The i-th random step is $$\ell_i=\pm \ell$$, with equal probability for the positive and negative sign, so that the average over time of the step is $$\langle \ell_i \rangle=0$$.

The position after N steps is
 * $$ X= \sum_i^N \ell_i$$

with an average
 * $$\langle X \rangle=\sum_i^N \langle \ell_i \rangle=0$$

meaning that on average the particle is around the same position. However the variance is
 * $$\langle X^2 \rangle=\langle (\sum_i^N \ell_i )^2 \rangle=\langle \sum_i^N  (\ell_i)^2 + 2\sum_{i<j} \ell_i \ell_j \rangle=N \ell^2$$

since $$\langle \ell_i^2 \rangle=\ell^2$$, and $$\langle \ell_i \ell_j \rangle=\langle \ell_i \rangle\langle \ell_j \rangle=0$$ for uncorrelated events.

Recalling that the number of step is linked to the total elapsed time by $$N=t/\Delta t$$, we have that $$\langle X^2 \rangle=D t$$, with $$D$$ a diffusion coefficient whose value is here $$D=\ell^2/\Delta t$$.

As a consequence a molecules explores a distance of standard deviation $$\sigma=\langle X^2 \rangle^{1/2} \sim (D t)^{1/2}$$.

Similarly an assembly of molecules (in a spot of dye for instance) spreads like $$\sigma \sim (D t)^{1/2}$$

Diffusion coefficient
In a liquid the value of the diffusion coefficient of a particule of radius $$R$$ given by the Stokes-Einstein formula.
 * $$D=\frac{kT}{6\pi \mu R}$$

The demonstration of this formula comes the solution for the motion of a particle with two main physical ingredients: a fluctuation velocity given by $$m\langle u^2 \rangle/2=kT/2$$, and a friction forces of value $$F=6\pi \mu R u$$.

Small molecules therefore diffuse faster

Table with values??

Many molecules are characterised by a concentration $$C(\mathbf{r},t)$$ which is a number per unit volume.